Rare Events in Financial Markets.pdf - Division of Applied Mathematics
Invariance in the Recurrence of Large Returns and the Validation of Models of Price Dynamics⇤
Lo-Bin Chang†, Stuart Geman‡, Fushing Hsieh§, and Chii-Ruey Hwang¶ March, 2013
⇤ keywords and phrases: market dynamics, distribution-free statistics, market time, GARCH, geometric Brownian motion † Department of Applied Mathematics, National Chiao Tung University ‡ Division of Applied Mathematics, Brown University § Department of Statistics, University of California, Davis ¶ Institute of Mathematics, Academia Sinica
Contents 1 Introduction
1
2 Waiting Times Between Large Returns 2.1 The role of the geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Empirical evidence for invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Connections to self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 4 5 7
3 Conditional inference, permutations, and hypothesis testing 3.1 Permutation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exploring time scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 10 13
4 Time Scale and Stochastic 4.1 Implied volatility . . . . 4.2 GARCH . . . . . . . . . 4.3 Market time . . . . . . .
13 13 14 16
Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Summary and concluding remarks
19
Abstract. Starting from a robust, nonparametric, definition of large returns (“excursions”), we study the statistics of their occurrences focusing on the recurrence process. The empirical waiting-time distribution between excursions is remarkably invariant to year, stock, and scale (return interval). This invariance is related to self-similarity of the marginal distributions of returns, but the excursion waiting-time distribution is a function of the entire return process and not just its univariate probabilities. GARCH models, markettime transformations based on volume or trades, and generalized (L´evy) random-walk models all fail to fit the statistical structure of excursions.
1
Introduction
Given a sequence of stock prices s0 , s1 , . . . recorded at fixed intervals, say every five minutes, let . rn = log snsn 1 , n = 1, 2, . . ., be the corresponding sequence of returns. Fix N and define an excursion to be a return that is large, in absolute value, relative to the set {r1 , r2 , . . . , rN }. Specifically, following Hsieh et al. (2012), define the excursion process, z1 , z2 , . . . , zN : ⇢ 1 if rn l or rn u zn = 0 if rn 2 (u, l) where l and u are, respectively, the 10’th and 90’th percentiles of {r1 , . . . , rN }. We call the event zn = 1 an excursion, since it represents a large movement of the stock relative to the chosen set of returns. We will study the distribution of waiting times between large stock returns by studying the distribution of the number of zeros between successive ones of the excursion process. Our motivation includes: 1. The empirical observation (cf. Chang et al., 2013) that this waiting-time distribution is nearly invariant to time scale (e.g. thirty-second, one-minute, or five-minute returns), to stock (e.g. IBM or Citigroup), and to year (e.g. 2001 or 2007). 1
2. The waiting-time to large returns is of obvious interest to investors, and much easier to study if, and to the extent that, it is invariant across time scale, stock, and year. 3. The particular waiting-time distribution found in the data and its invariance to time scale have implications for models of price and volatility movement. For instance, L´evy processes, “market-time” models based on volume or trades, and GARCH models are each one way or another inconsistent with the empirical data. 4. Overwhelmingly, the evidence for self-similarity comes from studies of the univariate (marginal) return distributions (e.g. evidence for a stable-law distribution), but marginal distributions leave data models underspecified. Waiting-time distributions provide additional, explicitly temporal, constraints, and these appear to be nearly universal. Larger returns can be studied by using more extreme percentiles. Although we have not experimented extensively, the empirical results we will report on appear to be qualitatively robust to the chosen percentiles and hence the definition of “large return.” In general, the upper and lower percentiles index a family of waiting-time distributions that might prove useful to systematically constrain the dynamics of price and volatility models. In §2, we study the invariance of the empirical waiting-time distribution. Starting with the L´evy type models, we first make a connection between the model-based distribution and the geometric distribution. To be concrete, let S(t) follow the “Black-Scholes model” (geometric Brownian motion) as an example: d log S(t) = µdt + dw(t), where w(t) is a standard Brownian motion. Because of the independent increments property of Brownian motion w(t), the return sequence under this model is exchangeable (i.e. the distribution of any permutation remains the same). Therefore, the empirical waiting-time distribution under this model is provably invariant to time scale and to time period. More specifically, the probability of getting a “large” return, with l =10’th percentile and u =90’th percentile, is exactly 0.2 at each return interval and the empirical waiting-time distribution is, therefore, nearly a geometric distribution with parameter 0.2 (see §2.1 for more detail). We emphasize the these considerations apply without modification not just to the geometric Brownian motion but to all of its popular generalizations as geometric L´evy processes. Not surprisingly (cf. “stochastic volatility”), the actual (i.e. empirical) waiting-time distribution is di↵erent from geometric. But what is surprising is the invariance of this distribution across time scale, stock, and year. In §2.2 we make an exhaustive comparison of empirical waiting-time distributions, using trading prices of approximately 300 stocks from the S&P 500 observed over the eight years from 2001 through 2008. Invariance to timescale is strong in all eight years; invariance to stock is strong in years 2001–2007 and less strong in 2008; and invariance across years is stronger for pairs of years that do not include 2008. (We have not studied the years since 2008.) In §2.3, we will connect waiting-time invariance to self-similarity, being careful to distinguish a self-similar process from a process having self-similar increments (i.e. distinguish dynamics from marginal distributions). Which of the state-of-the-art models of price dynamics are consistent with the empirical distribution of the excursion process? The existence of a nearly invariant waiting-time distribution between excursions provides a new tool for evaluating these models, through which questions of consistency with the data can be addressed using statistical measures of fit and hypothesis tests. In general, we will advocate for permutation and other combinatorial statistical approaches that robustly and efficiently exploit symmetries shared by large classes of models, supporting exact hypothesis tests as well as exploratory data analysis. In §3 we introduce some combinatorial tools for hypothesis testing and explore the implications of waiting-time distributions to the time scale of 2
volatility clustering. We continue with this approach, in §4, with a discussion of stochastic volatility modeling, as well as “market-time” and other stochastic time-change models. We conclude, in §5, with a summary and some proposals for price and volatility modeling.
2
Waiting Times Between Large Returns
There were 252 trading days in 2005. The traded prices of IBM stock (sn , n = 0, 1, . . . , 18,899) at every 5-minute interval from 9:40AM to 3:50PM (seventy five prices each day), throughout the 252 days, are plotted in Figure 1, Panel A.1 Often, activities near the opening and closing are not representative. To mitigate their influence, we exclude prices in the first ten minutes (9:30 to 9:40) . and last 10 minutes (3:50 to 4:00) of each day. The corresponding intra-day returns, rn = log snsn 1 , n = 1, 2, . . . , 18,648 (seventy four returns per day) are plotted in Panel B. Overnight returns are not included. 100
0.015
A
95
90
0.005
85
0
80
-0.005
75
-0.01
70
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
-0.015
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
0.015
0.015
C
0.01
0.005
0
0
-0.005
-0.005
-0.01
-0.01
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
D
0.01
0.005
-0.015
B
0.01
-0.015
0
20
40
60
80
100
120
140
160
180
200
Figure 1: Returns, percentiles, and the excursion process. A. IBM stock prices, every 5 minutes, during the 252 trading days in 2005. The opening (9:30 to 9:40) and closing (3:50 to 4:00) prices are excluded, leaving 75 prices per day (9:40,9:45,. . .,15:50). B. Intra-day 5-minute returns for the prices displayed in A. There are 252⇥74=18,648 data points. C. Returns, with the 10’th and 90’th percentiles superimposed. D. Zoomed portion of C with 200 returns. The “excursion process” is the discrete time zero-one process that signals (with ones) returns above or below the selected percentiles.
We declare a return “rare” if it is rare relative to the interval of study, in this case the calendar year 2005. We might, for instance, choose to study the largest and smallest returns in the interval, or the largest 10% and smallest 10%. Panel C shows the 2005 intra-day returns with the tenth and ninetieth percentiles superimposed. More generally, given any fractions f, g 2 [0, 1] (e.g. 0.1 and 1
The price at a specified time is defined to be the price of the most recent trade.
3
0.9), define lf = lf (r1 , . . . , rN ) = inf{r : #{n : rn r, 1 n N } ug = ug (r1 , . . . , rN ) = sup{r : #{n : rn r, 1 n N }
f · N} (1 g) · N }
(1) (2)
where, presently, N = 18,648. The lower and upper lines in Panel C are l.1 and u.9 , respectively. Panel D is a magnified view, covering r1001 , . . . , r1200 , but with l.1 and u.9 still figured as in equations (1) and (2) from the entire set of 18,648 returns.2 The excursion process is the zero-one process that signals large returns, meaning returns that either fall below lf or above ug : zn = 1rn lf or rn ug Hence zn = 1 for at least 20% of n 2 {1, 2, . . . , 18,648} in the Figure 1 example. Obviously, many generalizations are possible, involving indicators of single-tale excursions (e.g. f = 0, g = .9 or f = .1, g = 1) or many-valued excursion processes (e.g. zn is one if rn lf , two if rn ug , and zero otherwise). Or we could be more selective by choosing a smaller fraction f and a larger fraction g, and thereby move in the direction of truly rare events. (There is, then, an inevitable tradeo↵ between the magnitude of the excursions and the sample size; more rare events are studied at the cost of statistical power.) Here we will work with the special case f = .1 and g = .9, but a similar exploration could be made of these other excursion processes.
2.1
The role of the geometric distribution
As with the Black-Scholes model discussed in the introduction, any stochastic process with stationary and independent increments (i.e. any L´evy process) has exchangeable increments, and hence exchangeable returns if used as a model for the log-price distribution. What would the excursion waiting-time distribution look like under a geometric Brownian-motion model, or one of its generalizations to geometric L´evy? Specifically, assume d log S(t) = µdt + dw(t), where w(t) is a L´evy process. Then the return sequence Rk = log S(t0 + k t)
log S(t0 + (k
1) t), 8k = 1, 2, 3, · · · , n
(3)
is exchangeable. With the particular percentiles used here, the sequence z1 , z2 , . . . , zN has 20% 1’s and 80% 0’s. If real returns were exchangeable then the excursion process would be as well, since the percentiles lf and ug (equations 1 & 2) are symmetric functions of the returns. Hence, the probability that a 1 is followed immediately by another 1 (waiting time zero) is very nearly 0.2. (Not exactly 0.2, even ignoring edge e↵ects, because there are a finite number of 1’s – the first 1 of the pair uses one of them up.) The probability that exactly one 0 intervenes is very nearly (0.8)(0.2)=0.16, two 0’s very nearly (0.8)(0.8)(0.2)=0.128, and so-forth following the geometric distribution. In general, the waiting-time distribution for an exchangeable process converges to the geometric distribution as the number of excursions (number of return intervals) goes to infinity (Diaconis & 2
To break ties and to mitigate possible confounding e↵ects from “micro-structure,” prices are first perturbed, independently, by a random amount chosen uniformly from between ±$0.005.
4
Freedman, 1980, Chang et al., 2013). In this sense, the KS distance3 to the geometric distribution is a measure of departure of a return process from exchangeability, and can be used as a statistic to calibrate the temporal structure of real price data as well as proposed models of prices and returns (as will be discussed more deeply in §3 & §4). Figure 2 compares the empirical waiting-time distribution generated by 93,240 one-minute 2005 IBM returns to the geometric distribution with parameter 0.20. Obviously there is a substantial departure, characterized by high probabilities of short and long waits in the real data as compared to the geometric distribution. (The slope of the P-P curve is greater than one or less than one as waiting-time probabilities are respectively larger than or smaller than geometric.) Thus, for example, the empirical probability that the waiting time is zero (zn+1 = 1 given that zn = 1) is about 0.32 instead of 0.20. Indeed estimates of this probability reliably fall in a narrow range, from about 0.32 to 0.33, independent of the time interval with respect to which returns are defined, the stock from which the returns are derived, and the year from which the data is collected. In fact, the entire empirical waiting-time distribution is a near invariant to time scale, stock, and year, as we shall now demonstrate. Log probabilities of waiting time
IBM vs Geometric(0.2), KS=0.14457
0
1 IBM Geometric(0.2)
-2
0.9 0.8 0.7 0.6
-6
IBM
log probability
-4
-8
0.5 0.4 0.3
-10
0.2 -12 -14
0.1 0
10
20 30 waiting time
40
0
50
0
0.2
0.6 0.4 Geometric(0.2)
0.8
1
Figure 2: Geometric(0.2) and empirical waiting times. The empirical waiting-time distribution of 1-minute returns of IBM stock in 2005 was compared with the geometric distribution with parameter 0.2. Left panel: Log plots for the geometric distribution and the empirical waiting-time distribution. The x-axis is the waiting times and the y-axis is the log probabilities of the waiting times. Right Panel: P-P plots for the geometric distribution versus the empirical waiting-time distribution. The KS distance is the maximum horizontal (= maximum vertical) distance between the P-P curve (shown in blue) and the diagonal (shown in red).
2.2
Empirical evidence for invariance
Chang et al. (2013) and Hsieh et al. (2012) studied the waiting-time distribution between excursions, i.e. the distribution on the number of zeros between two ones. The empirical waiting-time 3 Given two cumulative distribution functions, F1 and F2 , the P-P plot is the two-dimensional curve from (0, 0) to (1, 1) defined by {(F1 (t), F2 (t) : t 2 R}. The Kolmogorov Smirnov (KS) distance is the maximum vertical (and horizontal) distance between the diagonal and the P-P plot, which is also the maximum distance between F1 and F2 :
dKS (F1 , F2 ) = sup |F1 (t) t
5
F2 (t)|
distribution from 2005, for the 18,648 5-minute returns, the 93,240 1-minute returns, and the 186,480 30-second returns of IBM are shown across the top of Figure 3. They are remarkably similar. 1 min
30 sec
2005ibm 30sec returns 0.35
0.3
0.3
0.3
0.25
0.25
0.25
0.35
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
40
30
50
60
70
80
0
100
90
0.8
0.6
0.6
0.6
5 min
0.8
5 min
0.8
0.4 0.2
0.2
0.2
0.4
0.6
0.8
1
0
30
50
40
60
70
80
90
100
30 sec vs 5 min KS=0.014
1
0.4
20
ibm2005 30 sec vs 5 min, KS=0.014186
1
0
10
ibm2005 1 min vs 5 min, KS=0.0052769
1
0
0
1 min vs 5 min KS=0.005
30 sec vs 1 min KS=0.016 ibm2005 30 sec vs 1 min, KS=0.015966
1 min
5 min ibm2005 5min returns
ibm2005 5min returns
0.35
0.4 0.2
0
0.2
0.6
0.4
0.8
1
0
0
0.2
1 min
30 sec
0.6
0.4
0.8
1
30 sec
Figure 3: Scale invariance. Top row: Empirical waiting-time distributions captured from 30-second, 1-minute, and 5-minute returns of IBM in 2005. Bottom row: P-P plots for the three waiting-time distributions taken two at a time, and their corresponding Kolmogorov-Smirnov distances.
Invariance to scale. The bottom row of Figure 3 has three P-P plots that come from taking the three waiting-time distributions (30-second, 1-minute, and 5-minute, shown in the top row) two at a time. The KS distances, one for each comparison, are also shown. The distribution of waiting times between excursions for IBM 2005 returns is strikingly invariant to the return interval. (We are using dKS here as a descriptive statistic, and not for the purpose of hypothesis testing. These waiting times are not precisely invariant, and many pairs that look well matched will nevertheless have small p-values, simply because of the large sample sizes.) Table 1: Scale invariance, aggregate data. Approximately 300 stocks were tested. Table shows median KS distances for pairwise comparisons of three time scales (30 seconds, 1 minute, 5 minutes) for each of the years 2001 through 2008.
year
2001
2002
2003
2004
2005
2006
30 sec vs 1 min
0.0199
0.0109
0.0148
0.0148
0.0163
0.0128
1 min vs 5 min
0.0253
0.0203
0.0197
0.017
0.0175
0.017
30 sec vs 5 min
0.0348
0.0247
0.0268
0.0259
0.0264
0.0223
2007
2008
0.0113 0.0103 0.0194
0.0143
0.0242 0.0172
The phenomenon is not unique to IBM, nor to the year 2005. We tested approximately 300 of 6
the S&P 500 stocks for the years 2001 through 2008. The results are summarized in Table 1. In this regard, 2008 is not an outlier, as can be seen from the last column of the table, and from the three histograms of KS distances, one for each pair of return intervals, over all stocks tested in 2008 (Figure 4). 30 secvsvs1-minute 1 min returns in 2008 IBM 30-second returns 80 60
1 min 5 min returns in 2008 IBM 1-minute returns vs vs 5-minute 50
30 sec vs min IBM 30-second returns vs 55-minute returns in 2008 50
40
40
30
30
20
20
10
10
40
20
0
0
0.05
0.1
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
Figure 4: Histogram of KS distances, 2008. Each panel shows the histogram of Kolmogorov-Smirnov distances between excursion waiting-time distributions at di↵erent time scales in 2008, for approximately 300 stocks.
As we will see shortly, self-similar processes have excursion waiting-time distributions that are invariant to scale. It is interesting, then, to note that the empirical evidence for waiting-time invariance is substantially weaker at larger intervals, e.g. using hourly or daily returns. This same progression is often observed in studies of self-similarity (cf. Mantegna and Stanley, 2000). Possibly, it can be traced to sample size. Because the return sequences are derived from a single calendar year, larger return intervals have smaller numbers of returns, and hence a larger variance of the empirical waiting-time distribution. For example, as a rough estimate, we can expect hourly returns to multiply the spread of a five-minute-return across-stock histogram of empirical KS distributions p (as in the lower-left panel of Figure 5) by about 60/5 ⇡ 3.5, which would substantially obscure the evidence for invariance. It is also possible that invariance systematically breaks down for larger return intervals. We have not explored either hypothesis. Invariance to stock and year. How do the excursion waiting-time distributions of one stock compare to those of another? For each of the eight years studied we compared the waiting-time distributions, for 5-minute returns, between all pairs of the 300 or so stocks in our data set. See Figure 5 and the accompanying table. With the possible exception of 2008, excursion waiting-time distributions are nearly invariant across stocks. Finally, we examined the change in waiting-time distributions from year to year. For each stock and each return interval (30-seconds, 1-minute, 5-minutes), we compared distributions between pairs of years. Table 2 indicates that waiting-time distributions were typically unchanged during the period 2001 to 2007, but considerably di↵erent during the financial crises of 2008.
2.3
Connections to self-similarity
Recall that P (t), t
0, is a self-similar process if there exists H L{P ( t), t
0} = L{
H
P (t), t
0 (“Hurst index”) such that 0}
for all 0, where L{Q(t), t 0} denotes the probability distribution (“law”) of the process Q(·). In other words, the joint distributions of (P ( t1 ), P ( t2 ), . . . , P ( tm )) and H (P (t1 ), P (t2 ), . . . , P (tm )) are the same, for all m, t1 , t2 , . . . , tm , and (e.g. Embrechts and Maejima, 2002). Let S(t), t 0, 7
IBM vs GPS in 2008, KS=0.059012 1
0.8
0.8
0.6
0.6
GPS
GPS
IBM vs GPS in 2005, KS=0.021442 1
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
IBM Histogram in 2005 6000
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
0
0.05
0.1
0.8
1
Histogram in 2008
6000
0
0.6
IBM
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
Year 2001 2002 2003 2004 2005 2006 2007 2008
Median 0.0264 0.0293 0.0228 0.0209 0.022 0.0208 0.0293 0.035
0.25
Figure 5: Invariance to stock. Comparisons of excursion waiting-time distributions for 5-minute returns between IBM and GPS in 2005 (top-left panel) and 2008 (top-right panel). Histograms of KS distributions for all pairs of stocks (bottom panels) show a breakdown of invariance across stocks in 2008 as compared to 2005. Table: Summary of year-by-year comparisons of waiting-time distributions across stocks. With the exception of 2008, waiting times are nearly invariant to stock.
be the price of a stock at time t. Beginning with Mandelbrot (1963,1967), it has often been observed that the marginal distribution of the (drift-corrected) increments in price, or more typically log price, is nearly self-similar, e.g. log S( t) log S( (t 1)) has nearly the same distribution as H H log S(t) log S(t 1), although di↵erent methods for estimating the exponent H give di↵erent values. Many authors (e.g. Calvet & Fisher, 2002 and Xu & Gencay, 2003) argued that the exponent is not constant (generally decreasing at larger scales) or that there are actually multiple exponents, as in the more general multi-fractal models. Within the framework of (single-exponent) self-similarity, the estimation method of Mantegna and Stanley (1995) is among the most convincing since it focuses on the centers of return distributions rather than their tails. Mantegna and Stanley Table 2: Year-to-year changes in excursion waiting-time distributions. Left column: Medians of KS distances, over all stocks and all pairs of years, 2001 through 2007. Right column: Median distances over all stocks from the single pair of years, 2005 and 2008. Waiting-time distributions in 2008 di↵er substantially from those of previous years.
year
2001 through 2007 2005 vs 2008
30-second returns
0.0236
0.0623
1- minute returns
0.0219
0.0681
5-minute returns
0.0228
0.0811
8
reported a Hurst index of about 0.71 for the S&P 500, with evidence for self-similarity spanning three orders of magnitude in the return interval, though as they and others (e.g. Bouchaud, 2001) pointed out, scaling breaks down at larger intervals. Additionally, many authors have studied empirical scaling through a variety of statistics that can be derived from, but are not directly equivalent to, self-similarity. For example, Gopikrishnan et al. (1999) investigated scaling properties of normalized returns, while Wang and Hui (2001) studied scaling phenomena using returns divided by their daily average returns. Gencay et al. (2001) explored wavelet variance, Matteo (2007) used R/S analysis, and Glattfelder et al. (2011) described 12 scaling laws in high-frequency FX data. Wang et al. (2006) studied the return interval between big volatilities and showed the persistence of scaling for a range of time resolution scales ( t = 1, 5, 10, 15, 30 min). Here we give a brief explanation of the mathematical relationship between self-similarity and scale invariance of the excursion waiting-distribution. Assume that the drift-corrected log price, P (·), is a self-similar process. Then, as for the return process, at scale with drift coefficient r, ( )
Rt ( )
) L{Rt , t
S( t) S( (t 1)) = P ( t) P ( (t 1)) + r . = log
1} = L{
H
= L{G
(1)
Rt + (
( )
(1) (Rt ), t
H
)r, t
1}
1} ( )
H where G( ) (x) is the monotone function H x + ( )r. Now let Zn , n = 1, 2, . . . , N , be the ex( ) cursion process corresponding to the return process Rn , n = 1, 2, . . . , N , for some scale (interval) (e.g. thirty seconds or five minutes). Since percentages are unchanged by monotone transfor( ) (1) mations, it follows that L{Zn , n = 1, 2, . . . , N } = L{Zn , n = 1, 2, . . . , N }, for all > 0. In short, self-similarity of the process P (t), t 0, implies that the excursion process, and therefore its waiting-time distribution, is invariant to scale. One family of self-similar models for P , made popular in finance by Mandelbrot’s 1963 paper, is the family of stable L´evy processes, i.e. the processes with stable, stationary, and independent ( ) ( ) increments. But the corresponding returns, R1 , R2 , . . ., are then iid for all > 0, and this violates volatility clustering. This shortcoming (already apparent to Mandelbrot in 1963) has led to the consideration of other self-similar models, that have stationary, and possibly stable, but notnecessarily-independent increments. One way to construct such processes is through random time changes of Brownian motion (Mandelbrot and Taylor, 1967, Clark, 1973, Anderson, 1996, Heyde, 1999, H. Geman et al., 2001). We will return to this approach in §4.3. A more direct approach is with fractional Brownian motion (FBM), which we will briefly discuss now as an illustration of the application of the excursion waiting-time distribution in the study of price fluctuations and their models. The FBMs are a family of self-similar Gaussian processes, one for each Hurst index H 2 (0, 1]. The particular value H = 1/2 is the ordinary Brownian motion. Which value of H best describes the 5-minute excursion waiting-time distribution of the 2005 IBM data? We explored di↵erent values of H. For each value, we generated 500 samples of the process P and extracted 18,648 returns, along with the corresponding excursion processes and their waiting-time distributions. (As discussed, in light of the fact that FBM is self-similar, the waiting-time distribution is invariant to .) Each waiting-time distribution has a KS distance to the distribution extracted from the real data. The
9
IBM2005 vs FBM with H=0.81, IBM vs FBM with H=0.81,KS=0.0461 KS=0.0461 1
0.13
0.9
0.12
0.8
0.11
0.7
0.1
0.6
FBM
Average distance KSK-S distance
Average distancebetween between IBM and Average K-S KS distance ibm-5min andFBM FBMgenerated generated data data 0.14
0.09
0.5
0.08
0.4
0.07
0.3
0.06
0.2
0.05
0.1
0.04 0.76
0.78
0.8
0.82 0.84 Hurst index
0.86
0.88
0.9
0
0
0.2
0.4
0.6
0.8
1
IBM
Figure 6: Fractional Brownian motion and excursion waiting times. Left panel: For each Hurst index H = 0.76, 0.77, . . . 0.89 we generated 500 FBM samples and extracted 18,648 returns, matching the 18,648 returns in the 5-minute 2005 IBM data. The average KS distances between the FBM excursion waiting times and the empirical IBM waiting times are plotted. The best fit, with KS about 0.046, is at H = 0.81. Right panel: P-P plot of excursion waiting-time distribution for IBM versus a sample from the best-fitting FBM. FBM overestimates the probabilities of short and very long waiting times.
averages of the 500 KS distances, for each of H = 0.76, 0.77, . . . , 0.89, are shown in the left-hand panel of Figure 6. The smallest KS distance over all examined H values was approximately 0.046, at H = 0.81. As can be seen from the right-hand panel of the figure, in comparison to real returns the fitted FBM model has too many short and too many long waiting times.
3
Conditional inference, permutations, and hypothesis testing
Our purpose in this section is to introduce some statistical tools that relate the near-invariance of the excursion waiting-time distribution to the temporal characteristics of the empirical return data, focusing particularly on the time scale of volatility clustering. In the following section, §4, these tools will be used to explore some familiar themes in price-dynamics modeling, including implied volatility, GARCH models, and various approaches to stochastic time change, a.k.a. market time. The statistical characterization of price and volatility fluctuations is obviously very complicated. Under the circumstance, model-free statistical methods can be particularly e↵ective tools for probing dynamics and discerning spatial and temporal patterns. The excursion process itself is an example, in that it avoids absolute thresholds and model-based parameter estimates. Permutation tests are another example, and are particularly suitable for relating the excursion process to the time scales operating in price fluctuations, as we shall now discuss.
3.1
Permutation tests
Returns are not exchangeable. If they were, there would be no stochastic volatility. Whereas we anticipated a failure of exchangeability, what is not apparent is the time scales involved in this departure of real dynamics from the basic random-walk models encapsulated by the geometric L´evy 10
processes. Are the five-minute returns of IBM locally exchangeable? What if we were to permute the 12 five-minute returns in each hour; would the price process look any di↵erent, either visually or statistically? As for visually, there is certainly no obvious “tell,” judging from a comparison of Panels B and C in Figure 7. Panel B plots the prices of IBM at five-minute intervals from 9:45AM to 3:45 PM, on a randomly selected day in 2005. Panel C plots a surrogate price sequence, derived from the original (i.e. the trajectory in Panel B) by permuting, randomly and independently, each set of twelve returns within each of the six hours. The surrogate sequence is started at the same price as the original and therefore again has the same price as the original at each ensuing hour. There is no visual clue that separates the real from the surrogate price sequence, and by our experience there never is one. How about statistically? Can we detect a di↵erence in the dynamics? Is there any indication that separates a real trajectory from its permutation surrogates? If so, how does this separation depend on time scale? We could as easily permute the set of five-minute returns within each week, each day, each hour, or each thirty-minute interval. At what time scale does exchangeability break down? Put di↵erently, at what time scales does volatility clustering operate? These questions can be systematically and robustly answered through a permutation test, and the resulting departure of the excursion waiting times between the permuted and original trajectories as measured through the KS distance. Let r1 , r2 , . . . , r18648 be the 18,648 five-minute intra-day returns, as defined in §2. Consider any statistic T (function of these returns), such as the KS distance between the excursion waiting-time distribution and the geometric distribution, as examined in Figure 7, Panel A. And consider the particular “null hypothesis,” Ho , that L{(R⇢(1) , R⇢(2) , . . . , R⇢(18648) )} is invariant to the permutations ⇢ in a set ⇧, where R1 , R2 , . . . , R18648 are the random variables associates with the observed returns. The point is not that we actually believe Ho (among other things, it violates volatility clustering), but rather that it leads to a measure of departure from exchangeability as determined by the particular statistic being examined, and the particular set of permutations ⇧. Under the null hypothesis a sequence of M iid permutations, ⇢1 (·), ⇢2 (·), . . . , ⇢M (·), chosen from the uniform distribution on the set of permutations in ⇧, produces a sequence of M + 1 conditionally iid T ’s, namely the observed Tobs = T (r1 , r2 , . . . , r18648 ) together with one additional value for each permutation: T⇢m = T (r⇢m (1) , r⇢m (2) , . . . , r⇢m (18648) ) m = 1, 2, . . . , M It follows that under Ho P r{#{m = 1, 2, . . . , M : T⇢m
Tobs }
N}
N +1 M +1
(4)
In other words, if N = #{m = 1, 2, . . . , M : T⇢m Tobs } then (N + 1)/(M + 1) is an exact p-value for Ho , in the direction of the alternative Ha that Tobs is larger than would be expected under Ho .4 Panel D of Figure 6 illustrates the test with M = 5,000 and ⇧ unrestricted, i.e. the entire permutation group on the sequence 1, 2, . . . , 18648. Since Tobs is larger than any of the values of T evaluated for the surrogate (i.e. permuted) sequences, N = 0 and the test has a p-value of 1 ⇡ 0.0002. As expected, the waiting-time distribution of real returns is not consistent with 5001 exchangeability, and in fact produced the largest deviation from geometric among all of the 5,001 sequences. Suppose now that we restrict ⇧ to include only local permutations, say within each day, or hour, or twenty-minute period. Then selecting from the uniform distribution on ⇧ is the 4 This is an instance of conditional inference, in that the test is conditioned on the particular realization. The correctness of the p-value follows from its correctness for any realization.
11
same thing as independently choosing a permutation for each (non-overlapping) day, or hour, or twenty-minute period, providing a mechanism for systematically exploring the time scale of volatility clustering. KS=0.131$
failure of exchangeability p
Lo-Bin Chang†, Stuart Geman‡, Fushing Hsieh§, and Chii-Ruey Hwang¶ March, 2013
⇤ keywords and phrases: market dynamics, distribution-free statistics, market time, GARCH, geometric Brownian motion † Department of Applied Mathematics, National Chiao Tung University ‡ Division of Applied Mathematics, Brown University § Department of Statistics, University of California, Davis ¶ Institute of Mathematics, Academia Sinica
Contents 1 Introduction
1
2 Waiting Times Between Large Returns 2.1 The role of the geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Empirical evidence for invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Connections to self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 4 5 7
3 Conditional inference, permutations, and hypothesis testing 3.1 Permutation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exploring time scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 10 13
4 Time Scale and Stochastic 4.1 Implied volatility . . . . 4.2 GARCH . . . . . . . . . 4.3 Market time . . . . . . .
13 13 14 16
Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Summary and concluding remarks
19
Abstract. Starting from a robust, nonparametric, definition of large returns (“excursions”), we study the statistics of their occurrences focusing on the recurrence process. The empirical waiting-time distribution between excursions is remarkably invariant to year, stock, and scale (return interval). This invariance is related to self-similarity of the marginal distributions of returns, but the excursion waiting-time distribution is a function of the entire return process and not just its univariate probabilities. GARCH models, markettime transformations based on volume or trades, and generalized (L´evy) random-walk models all fail to fit the statistical structure of excursions.
1
Introduction
Given a sequence of stock prices s0 , s1 , . . . recorded at fixed intervals, say every five minutes, let . rn = log snsn 1 , n = 1, 2, . . ., be the corresponding sequence of returns. Fix N and define an excursion to be a return that is large, in absolute value, relative to the set {r1 , r2 , . . . , rN }. Specifically, following Hsieh et al. (2012), define the excursion process, z1 , z2 , . . . , zN : ⇢ 1 if rn l or rn u zn = 0 if rn 2 (u, l) where l and u are, respectively, the 10’th and 90’th percentiles of {r1 , . . . , rN }. We call the event zn = 1 an excursion, since it represents a large movement of the stock relative to the chosen set of returns. We will study the distribution of waiting times between large stock returns by studying the distribution of the number of zeros between successive ones of the excursion process. Our motivation includes: 1. The empirical observation (cf. Chang et al., 2013) that this waiting-time distribution is nearly invariant to time scale (e.g. thirty-second, one-minute, or five-minute returns), to stock (e.g. IBM or Citigroup), and to year (e.g. 2001 or 2007). 1
2. The waiting-time to large returns is of obvious interest to investors, and much easier to study if, and to the extent that, it is invariant across time scale, stock, and year. 3. The particular waiting-time distribution found in the data and its invariance to time scale have implications for models of price and volatility movement. For instance, L´evy processes, “market-time” models based on volume or trades, and GARCH models are each one way or another inconsistent with the empirical data. 4. Overwhelmingly, the evidence for self-similarity comes from studies of the univariate (marginal) return distributions (e.g. evidence for a stable-law distribution), but marginal distributions leave data models underspecified. Waiting-time distributions provide additional, explicitly temporal, constraints, and these appear to be nearly universal. Larger returns can be studied by using more extreme percentiles. Although we have not experimented extensively, the empirical results we will report on appear to be qualitatively robust to the chosen percentiles and hence the definition of “large return.” In general, the upper and lower percentiles index a family of waiting-time distributions that might prove useful to systematically constrain the dynamics of price and volatility models. In §2, we study the invariance of the empirical waiting-time distribution. Starting with the L´evy type models, we first make a connection between the model-based distribution and the geometric distribution. To be concrete, let S(t) follow the “Black-Scholes model” (geometric Brownian motion) as an example: d log S(t) = µdt + dw(t), where w(t) is a standard Brownian motion. Because of the independent increments property of Brownian motion w(t), the return sequence under this model is exchangeable (i.e. the distribution of any permutation remains the same). Therefore, the empirical waiting-time distribution under this model is provably invariant to time scale and to time period. More specifically, the probability of getting a “large” return, with l =10’th percentile and u =90’th percentile, is exactly 0.2 at each return interval and the empirical waiting-time distribution is, therefore, nearly a geometric distribution with parameter 0.2 (see §2.1 for more detail). We emphasize the these considerations apply without modification not just to the geometric Brownian motion but to all of its popular generalizations as geometric L´evy processes. Not surprisingly (cf. “stochastic volatility”), the actual (i.e. empirical) waiting-time distribution is di↵erent from geometric. But what is surprising is the invariance of this distribution across time scale, stock, and year. In §2.2 we make an exhaustive comparison of empirical waiting-time distributions, using trading prices of approximately 300 stocks from the S&P 500 observed over the eight years from 2001 through 2008. Invariance to timescale is strong in all eight years; invariance to stock is strong in years 2001–2007 and less strong in 2008; and invariance across years is stronger for pairs of years that do not include 2008. (We have not studied the years since 2008.) In §2.3, we will connect waiting-time invariance to self-similarity, being careful to distinguish a self-similar process from a process having self-similar increments (i.e. distinguish dynamics from marginal distributions). Which of the state-of-the-art models of price dynamics are consistent with the empirical distribution of the excursion process? The existence of a nearly invariant waiting-time distribution between excursions provides a new tool for evaluating these models, through which questions of consistency with the data can be addressed using statistical measures of fit and hypothesis tests. In general, we will advocate for permutation and other combinatorial statistical approaches that robustly and efficiently exploit symmetries shared by large classes of models, supporting exact hypothesis tests as well as exploratory data analysis. In §3 we introduce some combinatorial tools for hypothesis testing and explore the implications of waiting-time distributions to the time scale of 2
volatility clustering. We continue with this approach, in §4, with a discussion of stochastic volatility modeling, as well as “market-time” and other stochastic time-change models. We conclude, in §5, with a summary and some proposals for price and volatility modeling.
2
Waiting Times Between Large Returns
There were 252 trading days in 2005. The traded prices of IBM stock (sn , n = 0, 1, . . . , 18,899) at every 5-minute interval from 9:40AM to 3:50PM (seventy five prices each day), throughout the 252 days, are plotted in Figure 1, Panel A.1 Often, activities near the opening and closing are not representative. To mitigate their influence, we exclude prices in the first ten minutes (9:30 to 9:40) . and last 10 minutes (3:50 to 4:00) of each day. The corresponding intra-day returns, rn = log snsn 1 , n = 1, 2, . . . , 18,648 (seventy four returns per day) are plotted in Panel B. Overnight returns are not included. 100
0.015
A
95
90
0.005
85
0
80
-0.005
75
-0.01
70
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
-0.015
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
0.015
0.015
C
0.01
0.005
0
0
-0.005
-0.005
-0.01
-0.01
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
D
0.01
0.005
-0.015
B
0.01
-0.015
0
20
40
60
80
100
120
140
160
180
200
Figure 1: Returns, percentiles, and the excursion process. A. IBM stock prices, every 5 minutes, during the 252 trading days in 2005. The opening (9:30 to 9:40) and closing (3:50 to 4:00) prices are excluded, leaving 75 prices per day (9:40,9:45,. . .,15:50). B. Intra-day 5-minute returns for the prices displayed in A. There are 252⇥74=18,648 data points. C. Returns, with the 10’th and 90’th percentiles superimposed. D. Zoomed portion of C with 200 returns. The “excursion process” is the discrete time zero-one process that signals (with ones) returns above or below the selected percentiles.
We declare a return “rare” if it is rare relative to the interval of study, in this case the calendar year 2005. We might, for instance, choose to study the largest and smallest returns in the interval, or the largest 10% and smallest 10%. Panel C shows the 2005 intra-day returns with the tenth and ninetieth percentiles superimposed. More generally, given any fractions f, g 2 [0, 1] (e.g. 0.1 and 1
The price at a specified time is defined to be the price of the most recent trade.
3
0.9), define lf = lf (r1 , . . . , rN ) = inf{r : #{n : rn r, 1 n N } ug = ug (r1 , . . . , rN ) = sup{r : #{n : rn r, 1 n N }
f · N} (1 g) · N }
(1) (2)
where, presently, N = 18,648. The lower and upper lines in Panel C are l.1 and u.9 , respectively. Panel D is a magnified view, covering r1001 , . . . , r1200 , but with l.1 and u.9 still figured as in equations (1) and (2) from the entire set of 18,648 returns.2 The excursion process is the zero-one process that signals large returns, meaning returns that either fall below lf or above ug : zn = 1rn lf or rn ug Hence zn = 1 for at least 20% of n 2 {1, 2, . . . , 18,648} in the Figure 1 example. Obviously, many generalizations are possible, involving indicators of single-tale excursions (e.g. f = 0, g = .9 or f = .1, g = 1) or many-valued excursion processes (e.g. zn is one if rn lf , two if rn ug , and zero otherwise). Or we could be more selective by choosing a smaller fraction f and a larger fraction g, and thereby move in the direction of truly rare events. (There is, then, an inevitable tradeo↵ between the magnitude of the excursions and the sample size; more rare events are studied at the cost of statistical power.) Here we will work with the special case f = .1 and g = .9, but a similar exploration could be made of these other excursion processes.
2.1
The role of the geometric distribution
As with the Black-Scholes model discussed in the introduction, any stochastic process with stationary and independent increments (i.e. any L´evy process) has exchangeable increments, and hence exchangeable returns if used as a model for the log-price distribution. What would the excursion waiting-time distribution look like under a geometric Brownian-motion model, or one of its generalizations to geometric L´evy? Specifically, assume d log S(t) = µdt + dw(t), where w(t) is a L´evy process. Then the return sequence Rk = log S(t0 + k t)
log S(t0 + (k
1) t), 8k = 1, 2, 3, · · · , n
(3)
is exchangeable. With the particular percentiles used here, the sequence z1 , z2 , . . . , zN has 20% 1’s and 80% 0’s. If real returns were exchangeable then the excursion process would be as well, since the percentiles lf and ug (equations 1 & 2) are symmetric functions of the returns. Hence, the probability that a 1 is followed immediately by another 1 (waiting time zero) is very nearly 0.2. (Not exactly 0.2, even ignoring edge e↵ects, because there are a finite number of 1’s – the first 1 of the pair uses one of them up.) The probability that exactly one 0 intervenes is very nearly (0.8)(0.2)=0.16, two 0’s very nearly (0.8)(0.8)(0.2)=0.128, and so-forth following the geometric distribution. In general, the waiting-time distribution for an exchangeable process converges to the geometric distribution as the number of excursions (number of return intervals) goes to infinity (Diaconis & 2
To break ties and to mitigate possible confounding e↵ects from “micro-structure,” prices are first perturbed, independently, by a random amount chosen uniformly from between ±$0.005.
4
Freedman, 1980, Chang et al., 2013). In this sense, the KS distance3 to the geometric distribution is a measure of departure of a return process from exchangeability, and can be used as a statistic to calibrate the temporal structure of real price data as well as proposed models of prices and returns (as will be discussed more deeply in §3 & §4). Figure 2 compares the empirical waiting-time distribution generated by 93,240 one-minute 2005 IBM returns to the geometric distribution with parameter 0.20. Obviously there is a substantial departure, characterized by high probabilities of short and long waits in the real data as compared to the geometric distribution. (The slope of the P-P curve is greater than one or less than one as waiting-time probabilities are respectively larger than or smaller than geometric.) Thus, for example, the empirical probability that the waiting time is zero (zn+1 = 1 given that zn = 1) is about 0.32 instead of 0.20. Indeed estimates of this probability reliably fall in a narrow range, from about 0.32 to 0.33, independent of the time interval with respect to which returns are defined, the stock from which the returns are derived, and the year from which the data is collected. In fact, the entire empirical waiting-time distribution is a near invariant to time scale, stock, and year, as we shall now demonstrate. Log probabilities of waiting time
IBM vs Geometric(0.2), KS=0.14457
0
1 IBM Geometric(0.2)
-2
0.9 0.8 0.7 0.6
-6
IBM
log probability
-4
-8
0.5 0.4 0.3
-10
0.2 -12 -14
0.1 0
10
20 30 waiting time
40
0
50
0
0.2
0.6 0.4 Geometric(0.2)
0.8
1
Figure 2: Geometric(0.2) and empirical waiting times. The empirical waiting-time distribution of 1-minute returns of IBM stock in 2005 was compared with the geometric distribution with parameter 0.2. Left panel: Log plots for the geometric distribution and the empirical waiting-time distribution. The x-axis is the waiting times and the y-axis is the log probabilities of the waiting times. Right Panel: P-P plots for the geometric distribution versus the empirical waiting-time distribution. The KS distance is the maximum horizontal (= maximum vertical) distance between the P-P curve (shown in blue) and the diagonal (shown in red).
2.2
Empirical evidence for invariance
Chang et al. (2013) and Hsieh et al. (2012) studied the waiting-time distribution between excursions, i.e. the distribution on the number of zeros between two ones. The empirical waiting-time 3 Given two cumulative distribution functions, F1 and F2 , the P-P plot is the two-dimensional curve from (0, 0) to (1, 1) defined by {(F1 (t), F2 (t) : t 2 R}. The Kolmogorov Smirnov (KS) distance is the maximum vertical (and horizontal) distance between the diagonal and the P-P plot, which is also the maximum distance between F1 and F2 :
dKS (F1 , F2 ) = sup |F1 (t) t
5
F2 (t)|
distribution from 2005, for the 18,648 5-minute returns, the 93,240 1-minute returns, and the 186,480 30-second returns of IBM are shown across the top of Figure 3. They are remarkably similar. 1 min
30 sec
2005ibm 30sec returns 0.35
0.3
0.3
0.3
0.25
0.25
0.25
0.35
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
40
30
50
60
70
80
0
100
90
0.8
0.6
0.6
0.6
5 min
0.8
5 min
0.8
0.4 0.2
0.2
0.2
0.4
0.6
0.8
1
0
30
50
40
60
70
80
90
100
30 sec vs 5 min KS=0.014
1
0.4
20
ibm2005 30 sec vs 5 min, KS=0.014186
1
0
10
ibm2005 1 min vs 5 min, KS=0.0052769
1
0
0
1 min vs 5 min KS=0.005
30 sec vs 1 min KS=0.016 ibm2005 30 sec vs 1 min, KS=0.015966
1 min
5 min ibm2005 5min returns
ibm2005 5min returns
0.35
0.4 0.2
0
0.2
0.6
0.4
0.8
1
0
0
0.2
1 min
30 sec
0.6
0.4
0.8
1
30 sec
Figure 3: Scale invariance. Top row: Empirical waiting-time distributions captured from 30-second, 1-minute, and 5-minute returns of IBM in 2005. Bottom row: P-P plots for the three waiting-time distributions taken two at a time, and their corresponding Kolmogorov-Smirnov distances.
Invariance to scale. The bottom row of Figure 3 has three P-P plots that come from taking the three waiting-time distributions (30-second, 1-minute, and 5-minute, shown in the top row) two at a time. The KS distances, one for each comparison, are also shown. The distribution of waiting times between excursions for IBM 2005 returns is strikingly invariant to the return interval. (We are using dKS here as a descriptive statistic, and not for the purpose of hypothesis testing. These waiting times are not precisely invariant, and many pairs that look well matched will nevertheless have small p-values, simply because of the large sample sizes.) Table 1: Scale invariance, aggregate data. Approximately 300 stocks were tested. Table shows median KS distances for pairwise comparisons of three time scales (30 seconds, 1 minute, 5 minutes) for each of the years 2001 through 2008.
year
2001
2002
2003
2004
2005
2006
30 sec vs 1 min
0.0199
0.0109
0.0148
0.0148
0.0163
0.0128
1 min vs 5 min
0.0253
0.0203
0.0197
0.017
0.0175
0.017
30 sec vs 5 min
0.0348
0.0247
0.0268
0.0259
0.0264
0.0223
2007
2008
0.0113 0.0103 0.0194
0.0143
0.0242 0.0172
The phenomenon is not unique to IBM, nor to the year 2005. We tested approximately 300 of 6
the S&P 500 stocks for the years 2001 through 2008. The results are summarized in Table 1. In this regard, 2008 is not an outlier, as can be seen from the last column of the table, and from the three histograms of KS distances, one for each pair of return intervals, over all stocks tested in 2008 (Figure 4). 30 secvsvs1-minute 1 min returns in 2008 IBM 30-second returns 80 60
1 min 5 min returns in 2008 IBM 1-minute returns vs vs 5-minute 50
30 sec vs min IBM 30-second returns vs 55-minute returns in 2008 50
40
40
30
30
20
20
10
10
40
20
0
0
0.05
0.1
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
Figure 4: Histogram of KS distances, 2008. Each panel shows the histogram of Kolmogorov-Smirnov distances between excursion waiting-time distributions at di↵erent time scales in 2008, for approximately 300 stocks.
As we will see shortly, self-similar processes have excursion waiting-time distributions that are invariant to scale. It is interesting, then, to note that the empirical evidence for waiting-time invariance is substantially weaker at larger intervals, e.g. using hourly or daily returns. This same progression is often observed in studies of self-similarity (cf. Mantegna and Stanley, 2000). Possibly, it can be traced to sample size. Because the return sequences are derived from a single calendar year, larger return intervals have smaller numbers of returns, and hence a larger variance of the empirical waiting-time distribution. For example, as a rough estimate, we can expect hourly returns to multiply the spread of a five-minute-return across-stock histogram of empirical KS distributions p (as in the lower-left panel of Figure 5) by about 60/5 ⇡ 3.5, which would substantially obscure the evidence for invariance. It is also possible that invariance systematically breaks down for larger return intervals. We have not explored either hypothesis. Invariance to stock and year. How do the excursion waiting-time distributions of one stock compare to those of another? For each of the eight years studied we compared the waiting-time distributions, for 5-minute returns, between all pairs of the 300 or so stocks in our data set. See Figure 5 and the accompanying table. With the possible exception of 2008, excursion waiting-time distributions are nearly invariant across stocks. Finally, we examined the change in waiting-time distributions from year to year. For each stock and each return interval (30-seconds, 1-minute, 5-minutes), we compared distributions between pairs of years. Table 2 indicates that waiting-time distributions were typically unchanged during the period 2001 to 2007, but considerably di↵erent during the financial crises of 2008.
2.3
Connections to self-similarity
Recall that P (t), t
0, is a self-similar process if there exists H L{P ( t), t
0} = L{
H
P (t), t
0 (“Hurst index”) such that 0}
for all 0, where L{Q(t), t 0} denotes the probability distribution (“law”) of the process Q(·). In other words, the joint distributions of (P ( t1 ), P ( t2 ), . . . , P ( tm )) and H (P (t1 ), P (t2 ), . . . , P (tm )) are the same, for all m, t1 , t2 , . . . , tm , and (e.g. Embrechts and Maejima, 2002). Let S(t), t 0, 7
IBM vs GPS in 2008, KS=0.059012 1
0.8
0.8
0.6
0.6
GPS
GPS
IBM vs GPS in 2005, KS=0.021442 1
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
IBM Histogram in 2005 6000
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
0
0.05
0.1
0.8
1
Histogram in 2008
6000
0
0.6
IBM
0.15
0.2
0.25
0
0
0.05
0.1
0.15
0.2
Year 2001 2002 2003 2004 2005 2006 2007 2008
Median 0.0264 0.0293 0.0228 0.0209 0.022 0.0208 0.0293 0.035
0.25
Figure 5: Invariance to stock. Comparisons of excursion waiting-time distributions for 5-minute returns between IBM and GPS in 2005 (top-left panel) and 2008 (top-right panel). Histograms of KS distributions for all pairs of stocks (bottom panels) show a breakdown of invariance across stocks in 2008 as compared to 2005. Table: Summary of year-by-year comparisons of waiting-time distributions across stocks. With the exception of 2008, waiting times are nearly invariant to stock.
be the price of a stock at time t. Beginning with Mandelbrot (1963,1967), it has often been observed that the marginal distribution of the (drift-corrected) increments in price, or more typically log price, is nearly self-similar, e.g. log S( t) log S( (t 1)) has nearly the same distribution as H H log S(t) log S(t 1), although di↵erent methods for estimating the exponent H give di↵erent values. Many authors (e.g. Calvet & Fisher, 2002 and Xu & Gencay, 2003) argued that the exponent is not constant (generally decreasing at larger scales) or that there are actually multiple exponents, as in the more general multi-fractal models. Within the framework of (single-exponent) self-similarity, the estimation method of Mantegna and Stanley (1995) is among the most convincing since it focuses on the centers of return distributions rather than their tails. Mantegna and Stanley Table 2: Year-to-year changes in excursion waiting-time distributions. Left column: Medians of KS distances, over all stocks and all pairs of years, 2001 through 2007. Right column: Median distances over all stocks from the single pair of years, 2005 and 2008. Waiting-time distributions in 2008 di↵er substantially from those of previous years.
year
2001 through 2007 2005 vs 2008
30-second returns
0.0236
0.0623
1- minute returns
0.0219
0.0681
5-minute returns
0.0228
0.0811
8
reported a Hurst index of about 0.71 for the S&P 500, with evidence for self-similarity spanning three orders of magnitude in the return interval, though as they and others (e.g. Bouchaud, 2001) pointed out, scaling breaks down at larger intervals. Additionally, many authors have studied empirical scaling through a variety of statistics that can be derived from, but are not directly equivalent to, self-similarity. For example, Gopikrishnan et al. (1999) investigated scaling properties of normalized returns, while Wang and Hui (2001) studied scaling phenomena using returns divided by their daily average returns. Gencay et al. (2001) explored wavelet variance, Matteo (2007) used R/S analysis, and Glattfelder et al. (2011) described 12 scaling laws in high-frequency FX data. Wang et al. (2006) studied the return interval between big volatilities and showed the persistence of scaling for a range of time resolution scales ( t = 1, 5, 10, 15, 30 min). Here we give a brief explanation of the mathematical relationship between self-similarity and scale invariance of the excursion waiting-distribution. Assume that the drift-corrected log price, P (·), is a self-similar process. Then, as for the return process, at scale with drift coefficient r, ( )
Rt ( )
) L{Rt , t
S( t) S( (t 1)) = P ( t) P ( (t 1)) + r . = log
1} = L{
H
= L{G
(1)
Rt + (
( )
(1) (Rt ), t
H
)r, t
1}
1} ( )
H where G( ) (x) is the monotone function H x + ( )r. Now let Zn , n = 1, 2, . . . , N , be the ex( ) cursion process corresponding to the return process Rn , n = 1, 2, . . . , N , for some scale (interval) (e.g. thirty seconds or five minutes). Since percentages are unchanged by monotone transfor( ) (1) mations, it follows that L{Zn , n = 1, 2, . . . , N } = L{Zn , n = 1, 2, . . . , N }, for all > 0. In short, self-similarity of the process P (t), t 0, implies that the excursion process, and therefore its waiting-time distribution, is invariant to scale. One family of self-similar models for P , made popular in finance by Mandelbrot’s 1963 paper, is the family of stable L´evy processes, i.e. the processes with stable, stationary, and independent ( ) ( ) increments. But the corresponding returns, R1 , R2 , . . ., are then iid for all > 0, and this violates volatility clustering. This shortcoming (already apparent to Mandelbrot in 1963) has led to the consideration of other self-similar models, that have stationary, and possibly stable, but notnecessarily-independent increments. One way to construct such processes is through random time changes of Brownian motion (Mandelbrot and Taylor, 1967, Clark, 1973, Anderson, 1996, Heyde, 1999, H. Geman et al., 2001). We will return to this approach in §4.3. A more direct approach is with fractional Brownian motion (FBM), which we will briefly discuss now as an illustration of the application of the excursion waiting-time distribution in the study of price fluctuations and their models. The FBMs are a family of self-similar Gaussian processes, one for each Hurst index H 2 (0, 1]. The particular value H = 1/2 is the ordinary Brownian motion. Which value of H best describes the 5-minute excursion waiting-time distribution of the 2005 IBM data? We explored di↵erent values of H. For each value, we generated 500 samples of the process P and extracted 18,648 returns, along with the corresponding excursion processes and their waiting-time distributions. (As discussed, in light of the fact that FBM is self-similar, the waiting-time distribution is invariant to .) Each waiting-time distribution has a KS distance to the distribution extracted from the real data. The
9
IBM2005 vs FBM with H=0.81, IBM vs FBM with H=0.81,KS=0.0461 KS=0.0461 1
0.13
0.9
0.12
0.8
0.11
0.7
0.1
0.6
FBM
Average distance KSK-S distance
Average distancebetween between IBM and Average K-S KS distance ibm-5min andFBM FBMgenerated generated data data 0.14
0.09
0.5
0.08
0.4
0.07
0.3
0.06
0.2
0.05
0.1
0.04 0.76
0.78
0.8
0.82 0.84 Hurst index
0.86
0.88
0.9
0
0
0.2
0.4
0.6
0.8
1
IBM
Figure 6: Fractional Brownian motion and excursion waiting times. Left panel: For each Hurst index H = 0.76, 0.77, . . . 0.89 we generated 500 FBM samples and extracted 18,648 returns, matching the 18,648 returns in the 5-minute 2005 IBM data. The average KS distances between the FBM excursion waiting times and the empirical IBM waiting times are plotted. The best fit, with KS about 0.046, is at H = 0.81. Right panel: P-P plot of excursion waiting-time distribution for IBM versus a sample from the best-fitting FBM. FBM overestimates the probabilities of short and very long waiting times.
averages of the 500 KS distances, for each of H = 0.76, 0.77, . . . , 0.89, are shown in the left-hand panel of Figure 6. The smallest KS distance over all examined H values was approximately 0.046, at H = 0.81. As can be seen from the right-hand panel of the figure, in comparison to real returns the fitted FBM model has too many short and too many long waiting times.
3
Conditional inference, permutations, and hypothesis testing
Our purpose in this section is to introduce some statistical tools that relate the near-invariance of the excursion waiting-time distribution to the temporal characteristics of the empirical return data, focusing particularly on the time scale of volatility clustering. In the following section, §4, these tools will be used to explore some familiar themes in price-dynamics modeling, including implied volatility, GARCH models, and various approaches to stochastic time change, a.k.a. market time. The statistical characterization of price and volatility fluctuations is obviously very complicated. Under the circumstance, model-free statistical methods can be particularly e↵ective tools for probing dynamics and discerning spatial and temporal patterns. The excursion process itself is an example, in that it avoids absolute thresholds and model-based parameter estimates. Permutation tests are another example, and are particularly suitable for relating the excursion process to the time scales operating in price fluctuations, as we shall now discuss.
3.1
Permutation tests
Returns are not exchangeable. If they were, there would be no stochastic volatility. Whereas we anticipated a failure of exchangeability, what is not apparent is the time scales involved in this departure of real dynamics from the basic random-walk models encapsulated by the geometric L´evy 10
processes. Are the five-minute returns of IBM locally exchangeable? What if we were to permute the 12 five-minute returns in each hour; would the price process look any di↵erent, either visually or statistically? As for visually, there is certainly no obvious “tell,” judging from a comparison of Panels B and C in Figure 7. Panel B plots the prices of IBM at five-minute intervals from 9:45AM to 3:45 PM, on a randomly selected day in 2005. Panel C plots a surrogate price sequence, derived from the original (i.e. the trajectory in Panel B) by permuting, randomly and independently, each set of twelve returns within each of the six hours. The surrogate sequence is started at the same price as the original and therefore again has the same price as the original at each ensuing hour. There is no visual clue that separates the real from the surrogate price sequence, and by our experience there never is one. How about statistically? Can we detect a di↵erence in the dynamics? Is there any indication that separates a real trajectory from its permutation surrogates? If so, how does this separation depend on time scale? We could as easily permute the set of five-minute returns within each week, each day, each hour, or each thirty-minute interval. At what time scale does exchangeability break down? Put di↵erently, at what time scales does volatility clustering operate? These questions can be systematically and robustly answered through a permutation test, and the resulting departure of the excursion waiting times between the permuted and original trajectories as measured through the KS distance. Let r1 , r2 , . . . , r18648 be the 18,648 five-minute intra-day returns, as defined in §2. Consider any statistic T (function of these returns), such as the KS distance between the excursion waiting-time distribution and the geometric distribution, as examined in Figure 7, Panel A. And consider the particular “null hypothesis,” Ho , that L{(R⇢(1) , R⇢(2) , . . . , R⇢(18648) )} is invariant to the permutations ⇢ in a set ⇧, where R1 , R2 , . . . , R18648 are the random variables associates with the observed returns. The point is not that we actually believe Ho (among other things, it violates volatility clustering), but rather that it leads to a measure of departure from exchangeability as determined by the particular statistic being examined, and the particular set of permutations ⇧. Under the null hypothesis a sequence of M iid permutations, ⇢1 (·), ⇢2 (·), . . . , ⇢M (·), chosen from the uniform distribution on the set of permutations in ⇧, produces a sequence of M + 1 conditionally iid T ’s, namely the observed Tobs = T (r1 , r2 , . . . , r18648 ) together with one additional value for each permutation: T⇢m = T (r⇢m (1) , r⇢m (2) , . . . , r⇢m (18648) ) m = 1, 2, . . . , M It follows that under Ho P r{#{m = 1, 2, . . . , M : T⇢m
Tobs }
N}
N +1 M +1
(4)
In other words, if N = #{m = 1, 2, . . . , M : T⇢m Tobs } then (N + 1)/(M + 1) is an exact p-value for Ho , in the direction of the alternative Ha that Tobs is larger than would be expected under Ho .4 Panel D of Figure 6 illustrates the test with M = 5,000 and ⇧ unrestricted, i.e. the entire permutation group on the sequence 1, 2, . . . , 18648. Since Tobs is larger than any of the values of T evaluated for the surrogate (i.e. permuted) sequences, N = 0 and the test has a p-value of 1 ⇡ 0.0002. As expected, the waiting-time distribution of real returns is not consistent with 5001 exchangeability, and in fact produced the largest deviation from geometric among all of the 5,001 sequences. Suppose now that we restrict ⇧ to include only local permutations, say within each day, or hour, or twenty-minute period. Then selecting from the uniform distribution on ⇧ is the 4 This is an instance of conditional inference, in that the test is conditioned on the particular realization. The correctness of the p-value follows from its correctness for any realization.
11
same thing as independently choosing a permutation for each (non-overlapping) day, or hour, or twenty-minute period, providing a mechanism for systematically exploring the time scale of volatility clustering. KS=0.131$
failure of exchangeability p
